Timeline for Small model categories?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 4, 2012 at 10:40 | vote | accept | Shlomi A | ||
| Oct 3, 2012 at 22:08 | comment | added | Shlomi A | Yes, I'm aware of it Todd. I've also just edited the question. Say, might there be some small sub-category of simplicial sets, for instance, that satisfies Quillen's original definition for a model category? | |
| Oct 3, 2012 at 21:54 | comment | added | Todd Trimble | Maybe you'd better edit your question then, Shlomi A. Of course, much of the literature these days uses all small limits and colimits, so you might consider whether Quillen's original definition is going to be the one you really want to work with. | |
| Oct 3, 2012 at 21:50 | history | edited | Shlomi A | CC BY-SA 3.0 |
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| Oct 3, 2012 at 21:46 | comment | added | Shlomi A | Alright. Fernando, by proposition V.2.3 (Freyd) in MacLane's Categories, a small complete category is a preorder. Therefore if I want to find a non-trivial example (or one which is not a preorder) then I better relax the axioms to Quillen's original definition, which required only finite (co)limits. Colin, the reference you've given looks neat. Thanks! :-) | |
| Oct 3, 2012 at 21:40 | answer | added | Zhen Lin | timeline score: 9 | |
| Oct 3, 2012 at 20:14 | answer | added | Todd Trimble | timeline score: 5 | |
| Oct 3, 2012 at 19:40 | comment | added | Fernando Muro | @Wouter, nice point, I wonder whether there's any model category structure there. @Colin, indeed, as you say, Quillen's 1967 original definition of model categories only asks for finite (co)limits, not a new discovery. | |
| Oct 3, 2012 at 19:29 | comment | added | Colin McQuillan | In arxiv.org/abs/1209.2699 Section 3.2 it is argued that it is reasonable to allow model categories to be only finitely (co)complete, and indeed this was in Quillen's original definition. | |
| Oct 3, 2012 at 19:28 | comment | added | Wouter Stekelenburg | @Muro: all complete lattices are complete categories, so now you do. | |
| Oct 3, 2012 at 19:23 | comment | added | Fernando Muro | I don't think so, model categories are (co)complete. Do you know many small (co)complete categories? | |
| Oct 3, 2012 at 19:22 | history | asked | Shlomi A | CC BY-SA 3.0 |